In case you weren't aware...in the Deep South, every public road that isn't a state highway or within a city/town falls under county jurisdiction.
I feel like that's true in most places (excluding stuff like National Forest Roads, etc.). Though I could be wrong I suppose. Though the difference is around where I've lived, all the roads are given traditional names, not route numbers.
Though, you may have some local variance as to how it's handled. For example, Ada County Idaho (home to Boise, Meridian, etc.) has one highway district, ACHD, which covers the whole county,
including city limits. ITD remains in control of state highways of course (though ACHD actually handles the signals on them). So the individual cities do not have direct control of their roads here.
Whereas neighboring Canyon County (home to Nampa, Caldwell, etc.) has 4 different county districts (not directly part of the county government as far as I can tell), plus each city is in charge of their own roads:

So if I'm counting right, Canyon County has 12 different agencies in charge of the roads (plus ITD for state highways).
EDIT: Then you have, probably the more common set-up, Deschutes County Oregon. The county directly maintains non-state/NF highways outside of the cities, rather than splitting it up into different districts that are not directly part of the county government.